If TU = x + 3, UV = 6, and TV = 2x - 5, what is TV?

We are given:

- \( TU = x + 3 \)
- \( UV = 6 \)
- \( TV = 2x - 5 \)

To find the value of \( TV \), we need to first find \( x \). Since \( TV \) is the total distance, we can use the fact that:

\[
TV = TU + UV
\]

Substitute the given values:

\[
2x - 5 = (x + 3) + 6
\]

Simplify the right-hand side:

\[
2x - 5 = x + 9
\]

Now, solve for \( x \):

1. Subtract \( x \) from both sides:

\[
x - 5 = 9
\]

2. Add 5 to both sides:

\[
x = 14
\]

Now, substitute \( x = 14 \) back into the expression for \( TV \):

\[
TV = 2x - 5 = 2(14) - 5 = 28 - 5 = 23
\]

So, \( TV = 23 \).

Would you like further details on any step? Here are some related questions you might find interesting:

1. What happens if \( TU \) is longer than \( UV \)?
2. How does the equation change if \( UV \) were not given?
3. Can you create a general formula for \( TV \) if only \( x \) and \( UV \) are known?
4. What is the physical interpretation of such equations in real-life distance problems?
5. How can you use substitution to solve similar systems of equations?

**Tip:** Always check if distances or lengths in geometry problems can be related through algebraic expressions!

Learn how to solve the equation for TV given the values TU = x + 3, UV = 6, and TV = 2x - 5. This step-by-step solution explains how to find x and use substitution to solve distance problems. Suitable for Grades 8-10.

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